For any sets A, B, C in a universe U:
Prove that:
If A n B = C n B and A n B' = C n B' then A = C
the "n" symbol means "intersect"
Hello,
$\displaystyle (A \cap B) \cup (A \cap B')=(C \cap B) \cup (C \cap B')$
$\displaystyle A \cap (B \cup B')=C \cap (B \cup B')$, by associative law (Algebra of sets - Wikipedia, the free encyclopedia)
But $\displaystyle B \cup B'=U$ by definition of the complement. And every set intersected with the universe will result in the set itself.
Hence $\displaystyle A=C$
This it true for $\displaystyle \left( {\forall X,Y} \right)\left[ {X = \left( {X \cap Y} \right) \cup \left( {X \cap Y'} \right)} \right]$.
So
$\displaystyle \begin{array}{rcl}
A & = & {\left( {A \cap B} \right) \cup \left( {A \cap B'} \right)} \\
{} & = & {\left( {C \cap B} \right) \cup \left( {C \cap B'} \right)} \\
{} & = & C \\
\end{array} $